Physics 220: Exercise on nonconservative forces
An object slides on a surface with coefficient of friction m _{k}. If it starts with speed v_{o}, and slides up a slope making angle q with the horizontal, how high does it go?
We start with a diagram:
Then we look at the energy of the system. We must consider gravitational potential energy and kinetic energy. Let's put the reference level at the bottom of the slope.
Before (bottom of slope) 
After (max height) 

Grav PE. 
0 
mgh 
Kinetic energy 
mv^{2}/2 
0 
Total mechanical energy 
mv^{2}/2 
mgh 
Now work done by nonconservative forces changes the TME:
W_{noncons }= E_{f} E_{I}
The only nonconservative force is friction, and the work done is:
since the frictional force and the displacement are in opposite directions.
Thus:
mgh mv^{2}/2 = fs
To proceed, we need f, which we can get from the friction relation f = m _{k}n and the freebody diagram:
f = m _{k}mgcosq
Finally, we use the geometry of the slope to relate h and s. Thus:
mgh mv^{2}/2 = fs =  m _{k}mgcosq h/sinq
The result reduces to v^{2}/2g when the coeffcient of friction is zero, or the angle is p /2, as expected. We also get zero when the angle is zero, which is also expected.